commutative

Let $S$ be a set and $\circ$ a binary operation on it. $\circ$ is said to be commutative if

$$a\circ b=b\circ a$$

for all $a,b\in S$.

Viewing $\circ$ as a function from $S\times S$ to $S$, the commutativity of $\circ$ can be notated as

$$\circ (a,b)=\circ (b,a).$$

Some common examples of commutative operations are

• addition over the integers: $m+n=m+n$ for all integers $m,n$

• multiplication over the integers: $m\cdot n=m\cdot n$ for all integers $m,n$

• addition over $n\times n$ matrices, $A+B=B+A$ for all $n\times n$ matrices $A,B$, and

• multiplication over the reals: $rs=sr$, for all real numbers $r,s$.

A binary operation that is not commutative is said to be non-commutative. A common example of a non-commutative operation is the subtraction over the integers (or more generally the real numbers). This means that, in general,

$$a-b\ne b-a.$$

For instance, $2-1=1\ne -1=1-2$.

Other examples of non-commutative binary operations can be found in the attachment below.

Remark. The notion of commutativity can be generalized to $n$-ary operations, where $n\ge 2$. An $n$-ary operation $f$ on a set $A$ is said to be commutative if

$$f(a_1,a_2,\mathrm{\dots },a_n)=f(a_{\pi (1)},a_{\pi (2)},\mathrm{\dots },a_{\pi (n)})$$

for every permutation $\pi$ on $\{1,2,\mathrm{\dots },n\}$, and for every choice of $n$ elements $a_i$ of $A$.

Title	commutative
Canonical name	Commutative
Date of creation	2013-03-22 12:22:45
Last modified on	2013-03-22 12:22:45
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	11
Author	CWoo (3771)
Entry type	Definition
Classification	msc 20-00
Synonym	commutativity
Synonym	commutative law
Related topic	Associative
Related topic	AbelianGroup2
Related topic	QuantumTopos
Related topic	NonCommutativeStructureAndOperation
Related topic	Subcommutative
Defines	non-commutative